Calculating the Force of Friction between surfaces but in a not so trivial context

Hi everyone: I need the brain of an engineer if anyone out there cares to help. I have a masters in Math from UofT but could use some knowledge from the smartest people- Engineers, I am now an experienced builder and yes everyone, we need smart people doing construction too- my math degree has facilitated my work as most people in construction lack brains...

I would like to know how engineers calculate the force of friction between surfaces but in a not so trivial context: I understand the logic of friction and calculating it when given a mass and a coefficient of static and kinetic friction, but my question is not so simple like OAC Physics I believe.

Q: how do we calculate friction considering surface area with two identical substances, in this case, slightly tapered CYLINDERS. consider this: I have one tapered cylinder inside another that is identical- like stacking buckets or plastic cups. i understand that if mass inside the top bucket increases by filling it with a liquid, it exerts lateral pressure on the sides of the elastic container and the container deforms slightly- this increases the horizontal force exerted on the container it is within/contained and therefore the frictional force between the two objects increases.

How is this force calculated? If i add water to a bucket and its within another bucket, how do we determine the increased frictional force between the two buckets? I think its a function of contact surface area of the objects with the mass of the substance added (uniformly distributed in the bucket) but i am not an engineer.... If anyone can tell me how to calculate this, even roughly, it would be greatly appreciated- using math would be helpful as i am quite good at it.

I have a feeling this is a quite complicated to calculate accurately and many factors modify the result but if someone can educate me with a basic way to approximate such a force, it would be great as I am just trying to understand how frictional forces are effected when changing the external surface area contact? would two very tall and small diameter buckets result in less frictional force than two very short and wide buckets stacked?

The volume is constant. I am trying to minimize the friction here and just as we can maximize the volume of a constructed shape given a fixed surface area of material to make the shape, I believe we can also maximize or minimize friction as well but i don't know for certain. To me the horizontal cross section of the cylinder at any given height specifically, and the corresponding lateral force exerted at that height, will be a function of the pressure exerted with gravity on it from the mass above, thereby making the relative lateral force given a height in the container, differential- I am not sure....? I just need know how to change the shape of my object so that when filled and stacked in another identical (but empty) object, the friction between the objects is minimized.

If anyone can answer or direct me to finding one of utility, its sincerely appreciated! and If the answer directs me to my ultimate result which i believe is possible, they will receive a lot more and this is seriously my intent- If someone helps me here and I use that help to bring my idea to reality, they will be very happy.

I don't entirely understand the later part of the problem statement . You can reduce friction torque to zero by maintaining a clearance between inner and outer buckets .

If there must be some contact then you would need to look at several different configurations and find an optimal one .

Agree, and I see this done (and not done) in practice with ~ 5 Gallon plastic buckets. In some cases, there is an outer lip near the top of the bucket that limits how far the top bucket will drop into the lower one. If this provides enough clearance to allow for the top bucket to be filled, but not deform enough to touch the sides of the lower bucket, there is no friction from the sides.

When not done, it can be a bear to separate the two buckets.

So a combination of defining where the lip needs to be for a given clearance, and designing the bucket for a given deformation when filled that is less than that clearance should get you near zero friction.

Alternately, it might help to provide one or more inlet(s) on the bottom bucket/sloped-cylinder, and pump air or liquid into the space. That pressure should help separate the two.

There are plenty of similar friction problems featuring masses on slopes or wedges and the solution to this question will be similar. Just calculate the normal force between the two tapered cylinders. Its probably just something like

In the most common model of friction (F=μN) the friction force is independent of contact area (If the area is increased the pressure reduces and these effects are usually considered to cancel out). However this is only a model of how friction works, it's not derived from first principles. Other models may be more accurate in some cases - for example it's widely believed that reducing the pressure in car tyres gives you more grip on snow because it increases the contact area. If you ever need reliable results from a friction problem you have to do experiments. Airports usually measure runway friction, I don't think they try and calculate it.